This is the first in a series of informal discussions about Grassmannians and flag varieties, with the goal of describing both how to represent Schubert problems on a computer and some goals of this computational study.

An outline of the discussions as they are given is at https://franksottile.github.io/talks/25/SchubertProblems.txt

This continues my informal discussion about Schubert problems in the Grassmannian. After describing Schubert problems, I will pivot to discuss some basics of numerical homotopy continuation, in preparation for the visit of Thomas Yahl in two weeks.

(This is a reprise of the talk in the MPHA seminar on Friday, March 7.)
To a Z^d-periodic weighted graph with vertices V, we may associated a periodic graph operator that acts on ℓ₂(V). After Floquet transform, we obtain its Bloch variety, which is an algebraic hypersurface in T^d×R whose projection to R is the spectrum of the operator. Features on Bloch varieties such as Dirac (double) points, critical points, and their level sets (Fermi varieties) reflect spectral properties of the operator.
    With students Faust and Robinson, and using the Brazos cluster, we investigated over 2.1 million small periodic graphs, recording invariants and features of their Bloch and Fermi varieties. In this talk, I will briefly discuss the background and present some examples of interesting behavior of Bloch varieties that we uncovered.